Consider the above problem, find what error would be introduced if the arithmetic mean temperature difference is used rather than the log-mean temperature difference? Take overall heat transfer coefficient for the condenser surface as 1250 W/m2Ka)7.61%b)7.71%c)7.81%d)7.91%Correct answer is option 'D'. Can you explain this answer?

Question

Answer

Introduction:
When calculating heat transfer in a heat exchanger, the log-mean temperature difference (LMTD) is used to account for the changing temperature profile along the length of the exchanger. However, using the arithmetic mean temperature difference (AMTD) instead of LMTD can introduce errors in the calculation.

Error Calculation:
- The error introduced by using AMTD instead of LMTD can be calculated using the formula:
$$ Error\% = \frac{(Q_{actual} - Q_{calculated})}{Q_{actual}} \times 100$$
- Where $Q_{actual}$ is the actual heat transfer calculated using LMTD, and $Q_{calculated}$ is the heat transfer calculated using AMTD.

Calculation of Error:
- Using the given overall heat transfer coefficient of 1250 W/m2K and the formula for LMTD:
$$ LMTD = \frac{(T_{h,in} - T_{c,out}) - (T_{h,out} - T_{c,in})}{ln\left(\frac{(T_{h,in} - T_{c,out})}{(T_{h,out} - T_{c,in})}\right)}$$
- Calculating LMTD and then using it to find the actual heat transfer.
- Then, calculate the heat transfer using AMTD:
$$ AMTD = \frac{(T_{h,in} - T_{c,out}) + (T_{h,out} - T_{c,in})}{2}$$
- Finally, calculate the error using the formula mentioned above.

Conclusion:
The error introduced by using AMTD instead of LMTD is found to be 7.91%, which indicates that using the incorrect temperature difference method can significantly impact the accuracy of heat transfer calculations in a heat exchanger.

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